Some preliminaries: In order for there to be a fallacy, one of these ideas should be intended to serve as support for the other. So, presumably the conclusion (or supported idea) has something to do with spending time and money on luxuries. And that idea isn't presumably about believing. I take it that the most plausible reading is that enjoying luxuries is wrong while bad things are happening.
If your thought is that the two ideas don't have anything to do with one another, then offering the one as putative support for the other is a non sequitur (from Latin: "It does not follow."). The one does not support the other. I think that's the fallacy you're looking for.
But as with most fallacies, there are times when it's wrong and times when it's right, and I think it is not completely obvious which situation we're in.
Certainly, bad things happening can't itself make enjoying luxuries wrong without a further idea to connect them. But I think there are charitable and likely interpretations of the speaker's intent that fill in that link. The tacit (or unstated) premises might be (a) that the time and money spent on luxuries could be spent instead on stopping bad things, and (b) that it is important (or right, or our duty, or just good) to stop bad things that are happening.
If that's plausible, you could insist that the argument is a non sequitur, but now to do so you would have to deny one of three claims:
- that the named bad things are actually bad
- that it is possible to divert time and money spent on luxuries to addressing bad things
- that it is important, right, or just good to stop bad things from happening
You imply that you sometimes reject (1) in such arguments. I would find it hard to agree, though, that at least poverty and sickness are not bad. (Anger, meanwhile, is often just fine.)
But if you accept that the named thing is actually bad, then if you want to assert that there's a fallacy at work, you need to deny (2) or (3). In some cases of sickness, (2) does not apply, because no amount of time or money can fix them. That might even apply to some cases of poverty. But there are clearly many cases of poverty and sickness to which (2) does apply. The cholera outbreak in Haiti after their 2010 earthquake, or the current situation in the Philippines after their typhoon involve a lot of people suffering who don't have the resources to fix their problems. Meanwhile, it is possible to reach them with meaningful, economically-efficient aid. And so, in that sense, time and money can work to address bad situations.
So, it might be that you'd just want to deny (3). You don't think it's good or important to stop bad things — or rather, you dont think that stopping bad things is as great a good, or is as important, as your enjoyment of luxuries.
That yields a consistent position, though it's one that ethicists argue against in a number of ways. More specifically, philosopher Peter Singer argues that many of us do accept (3), and even demand that others accept it, when faced with a situation like a nearby drowning child, but deny it for more distant situations we would rather not think about. Singer argues, among other things, that distance is not morally relevant, nor is the presence of others who are able to stop the bad things but don't. His argument is worth engaging.
So yes, you can assert that a fallacy is involved, by noting that the conclusion only follows when other tacit or silent premises are presumed, and then by denying the plausibility of those. Whether or not you categorically deny them is, of course, beyond the scope of your question.